NONLINEAR VIBRATIONS OF AXIALLY ACCELERATING VISCOELASTIC TIMOSHENKO BEAMS

研究了轴向加速黏弹性Timoshenko梁的非线性参数振动。参数激励是由径向变化张力和轴向速度波动引起的。引入了取决于轴向加速度的径向变化张力,同时还考虑了有限支撑刚度对张力的影响。应用广义哈密尔顿原理建立了Timoshenko梁耦合平面运动的控制方程和相关的边界条件。黏弹性本构关系采用Kelvin模型并引入物质时间导数。耦合方程简化为具有随时间和空间变化系数的积分-偏微分型非线性方程。采用直接多尺度法分析了Timoshenko梁的组合参数共振。根据可解性条件得到了Timoshenko梁的稳态响应,并应用Routh-Hurvitz判据确定了稳态响应的稳定性。最后通过一系列数值例子描述了黏弹性系数、平均轴向速度、剪切变形系数、转动惯量系数、速度脉动幅值、有限支撑刚度参数以及非线性系数对稳态响应的影响。     

Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions

This work explores the steady-state periodic transverse responses with their stabilities of axially accelerating viscoelastic strings. Longitudinally varying tension due to the axial acceleration is recognized in the modeling, while the tension was approximatively assumed to be longitudinally uniform in previous investigations. Exact internal resonances are highlighted in the analysis, while the resonances have been neglected in all available works. A governing equation of transverse nonlinear vibration is derived from the generalized Hamilton principle and the Kelvin viscoelastic model on the assumption that the string deformation is not infinitesimal, but still small. The axial speed is supposed to be a small simple harmonic fluctuation about the constant mean axial speed. The method of multiple scales is applied to solve the governing equation in the parametric resonances when the axial speed fluctuation frequency approaches the first three natural frequencies of the linear generating system based on 1–3 term truncations. The amplitude, the existence conditions, and the stability are determined, and the effects of the viscosity, the mean axial speed, the axial speed fluctuation amplitude, and the axial support rigidity on the amplitude and the existence are examined via the numerical examples. It is found that the 1-term, the 2-term, and the 3-term truncations yield the qualitatively same and the quantitatively close results in the case that there exist the exact internal resonances among the first three frequencies.     

Vibration and stability of an axially moving beam immersed in fluid

Vibration and stability are investigated for an axially moving beam in fluid and constrained by simple supports with torsion springs. The equations of motion of the beam with uniform circular cross-section, moving axially in a horizontal plane at a known rate while immersed in an incompressible fluid are derived first. An “axial added mass coefficient” and an initial tension are implemented in these equations. Based on the Differential Quadrature Method (DQM), a solution for natural frequency is obtained and numerical results are presented. The effects of axially moving speed, axial added mass coefficient, and several other system parameters on the dynamics and instability of the beam are discussed. Particularly, natural frequency in terms of the moving speed is presented for fixed–fixed, hinged–hinged and hybrid supports with torsion spring. It is shown that when the moving speed exceeds a certain value, the beam becomes subject to buckling-type instability. The variations of the lowest critical moving speed with several key parameters are also investigated.     

A geometrically exact approach to the overall dynamics of elastic rotating blades—part 2: flapping nonlinear normal modes

The geometrically exact equations of motion about the prestressed state discussed in part 1 (i.e., the nonlinear equilibrium under centrifugal forces) are expanded in the Taylor series of the incremental displacements and rotations to obtain the third-order perturbed form. The expanded form is amenable to a perturbation treatment to unfold the nonlinear features of free undamped flapping dynamics. The method of multiple scales is thus applied directly to the partial-differential equations of motion to construct the backbone curves of the flapping modes and their nonlinear approximations when they are away from internal resonances with other modes. The effective nonlinearity coefficients of the lowest three flapping modes of elastic isotropic blades are investigated when the angular speed is changed from low- to high-speed regimes. The novelty of the current findings is in the fact that the nonlinearity of the flapping modes is shown to depend critically on the angular speed since it can switch from hardening to softening and vice versa at certain speeds. The asymptotic results are compared with previous literature results. Moreover, 2:1 internal resonances between flapping and axial modes are exhibited as singularities of the effective nonlinearity coefficients. These nonlinear interactions can entail fundamental changes in the blade local and global dynamics.     

非对称混杂边界轴向运动Timoshenko梁橫向振动分析

研究两端带有扭转弹簧且弹簧系数均可任意变化的非对称混杂边界下的轴向运动Timoshenko梁的横向振动.利用非对称混杂边界条件推导对应任意弹簧系数的系统超越方程以及特征函数.运用数值方法计算系统的固有频率及其相应的模态函数,并研究确定梁的刚度、轴向速度以及边界处扭转弹簧的刚度的影响.通过数值算例,比较7imoshenko梁、瑞利梁、剪切梁和欧拉梁的固有频率随轴向速度的变化,分析转动惯量和剪切变形的影响.     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

     

Resonances of an in-extensional asymmetrical spinning shaft with speed fluctuations

In this study, main and parametric resonances of an asymmetrical spinning shaft with in-extensional nonlinearity and large amplitude are simultaneously investigated. The main resonance is due to inhomogeneous part of the equations of motion, which is due to dynamic imbalances of shaft whereas the parametric resonances are due to parametric excitations due to speed fluctuations and a shaft asymmetry. The shaft is simply supported with unequal mass moments of inertia and flexural rigidities in the direction of principal axes. The equations of motion are derived by the extended Hamilton principle. The stability and bifurcations are obtained by multiple scales method, which is applied to both partial and ordinary differential equations of motion. The influences of asymmetry of shaft, speed fluctuations, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation are studied. To investigate the effect of speed fluctuations on the bifurcations and stability the loci of bifurcation points are plotted as function of damping coefficient. The numerical solutions are used to verify the results of multiple scales method. The results of multiple scales method show a good agreement with those of numerical solutions.     

Nonlinear dynamics of a planar beam–spring system: analytical and numerical approaches

The multiple timescales method is applied to the exact partial differential equations of the planar motion of a hinged–simply supported beam with a linear axial spring of arbitrary stiffness. The forced-damped and free oscillations of the system around frequencies corresponding to nth natural bending mode are examined thoroughly and compared with numerical simulations as well as with already published results obtained by Lindstedt–Poincaré method. A special numerical technique using explicit finite element method to draw the frequency–response curves is appositely developed. The well-known jump phenomena between resonant and non-resonant branches, as well as superharmonic resonances, have been detected numerically.     

风作用下覆冰悬索的非平面非线性动力学

主要研究侧向风载荷作用下小垂度覆冰悬索的非线性非平面运动的复杂动力学.根据分析力学、弹性力学和空气动力学理论,建立覆冰悬索3个自由度非线性振动的偏微分运动方程,并对其进行无量纲化,运用Galerkin方法对偏微分运动方程进行离散得到3个自由度的常微分方程,再利用多尺度法得到面内主共振2:1内共振的平均方程.利用数值方法研究悬索的非线性运动,结果表明系统呈现周期、多倍周期、概周期和混沌运动的规律.     

Stability and bifurcations of an axially moving beam with an intermediate spring support

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton??s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.     

High-dimensional time series prediction using kernel-based Koopman mode regression

This paper investigates the nonlinear dynamics of a doubly clamped piezoelectric nanobeam subjected to a combined AC and DC loadings in the presence of three-to-one internal resonance. Surface effects are taken into account in the governing equation of motion to incorporate the associated size effects at nanoscales. The reduced-order model equation (ROM) is obtained based on the Galerkin method. The multiple scales method is applied directly to the nonlinear equation of motion and associated boundary conditions to obtain the modulation equations. The equilibrium solutions of the modulation equations and the dynamic solutions of the ROM equation are investigated in the case of primary and principal parametric resonances of the first mode. Stability, bifurcations and frequency response curves of the nanobeam are investigated. Dynamic behaviors of the motion are shown in the form of time traces, phase portraits, Poincare sections and fast Fourier transforms. The results indicate rich dynamic behaviors such as Hopf bifurcations, periodic and quasiperiodic motions in both directly and indirectly excited modes illustrating the influence of modal interactions on the response.     

Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations

Stability is investigated for an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation, not simply the partial time derivative. The method of multiple scales is applied directly to the governing equation without discretization. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams constrained by simple supports with rotational springs in parametric resonance. The finite difference schemes are developed to solve numerically the equation of axially accelerating viscoelastic beams with fixed supports for the instability regions in the principal parametric resonance. The numerical calculations confirm the analytical results. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.     

Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam

This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.     

Large Amplitude Three-Dimensional Free Vibrations of Inclined Sagged Elastic Cables

The nonlinear characteristics in the large amplitude three-dimensionalfree vibrations of inclined sagged elastic cables are investigated. Amodel formulation which is not limited to cables having smallsag-to-span ratios and takes into account the axial deformation effectis considered. Based on a multi-degree-of-freedom cable model, a finitedifference discretization is employed within a numerical solution of thegoverning equations of three-dimensional coupled motion. Variousnumerical examples of arbitrarily inclined sagged cables with initialout-of-plane or in-plane motions are carried out for the case of aspecified end tension. The major findings consist of highlighting theextent of two-and three-dimensional nonlinear couplings, the occurrenceof nonlinear dynamic tensions, and the meaningfulness of modaltransition phenomena ensuing from the activation of various internalresonance conditions. The influence of cable inclination on thenonlinear dynamic behavior is also evaluated. Comprehensive discussionand comparison of large amplitude free vibrations of horizontal andinclined sagged cables are presented.     

Bifurcations and chaotic dynamics in suspended cables under simultaneous parametric and external excitations

The bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degree-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary resonant excitation, and principal parametric excitation of suspended cables is considered. Using the method of multiple scales, a parametrically and externally excited system is transformed to the averaged equations. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, and Hopf bifurcations. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Five branches of dynamic solutions are found. Three of these branches that emerge from two Hopf bifurcations and the other two are isolated. The two Hopf bifurcation points, one is supercritical Hopf bifurcation point and another is primary Hopf bifurcation point. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging, boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, homoclinic explosions and hyperchaos are also observed.     

The control of a permanent rotational motion and equilibrium position of a spacecraft

This article has adopted an analytical method to obtain a non-linear control law to reach the exponential asymptotic stablity of the permanent rotational motion of a spacecraft. The control moments achieving this rotational motion are obtained. The control moments to establish exponential asymptotic stablity of the mentioned motion are obtained as non-linear functions of the phase coordinates of the spacecraft. The general solution of the equations of perturbed motion is derived. Furthermore, analysis and numerical simulation study of this solution are presented. For numerical examples the time needed for control is calculated. An equilibrium position of the spacecraft is proved to be exponentially asymptotically stable as a special case of the above-studied problem.     

Subharmonics analysis of nonlinear flexural vibrations of piezoelectrically actuated microcantilevers

Using the method of multiple scales, an extensive frequency response and subharmonic resonance analysis of the equations of motion governing the nonlinear flexural vibrations of piezoelectrically actuated microcantilevers is performed. Such comprehensive understanding of the nonlinear response and subharmonics analysis of these microcantilevers is, indeed, justified by the applications of piezoelectrically actuated microcantilevers that are increasingly becoming popular in many science and engineering areas including scanning force microscopy, biosensors, and microactuators. Along this line, the method of multiple scales is used to derive the 2× and 3× subharmonic resonances appearing in nonlinear flexural vibrations of a piezoelectrically actuated microcantilever. An experimental examination is performed in order to verify the analytical results. The analytical and experimental results yield the same system response for the fundamental frequency. In addition, the experimental results demonstrate the presence of subharmonic resonances that are supported by numerical simulations of the equations of motion. The experimental mode shapes of these subharmonic frequencies are also measured and compared with fundamental frequency.     

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electro- magnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period- doubling motion and quasi-period motion are discussed.     

Whirling of a forced cantilevered beam with static deflection. I: Primary resonance

The response of a slender, clastic, cantilevered beam to a transverse, vertical, harmonic excitation is investigated. The effects of nonlinear curvature, nonlinear inertia, viscous damping and static load are included. Previous work often has neglected the static deflection caused by the weight of the beam, which adds quadratic terms in the governing equations of motion. Galerkin's method is used with three modes and approximate solutions of the temporal equations are obtained by the method of multiple scales. Primary resonance is treated here, and out-of-plane motion is possible in the first and second modes when the principal moments of inertia of the beam cross-section are approximately equal. In Parts II and III, secondary resonances and nonstationary passages through various resonances are considered.     

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton??s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincaré maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincaré sections.